A proof of snevily’s conjecture

We prove Snevily’s conjecture, which states that for any positive integer k and any two k-element subsets {a ₁, …, a _k } and {b ₁, …, b _k } of a finite abelian group of odd order there exists a permutation π ∈ S _k such that all sums a _i + b _π(i) (i ∈ [1, k]) are pairwise distinct.     

Exterior algebras and two conjectures on finite abelian groups

Let G be a finite abelian group with |G| > 1. Let a ₁, …, a _k be k distinct elements of G and let b ₁, …, b _k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation π on {1, …,k} such that a ₁ b _π(1), …, a _k b _π(k) are distinct, provided that any other prime divisor of |G| (if there is any) is greater than k!. This in particular confirms the Dasgupta-Károlyi-Serra-Szegedy conjecture for abelian p-groups. We also pose a new conjecture involving determinants and characters, and show that its validity implies Snevily’s conjecture for abelian groups of odd order. Our methods involve exterior algebras and characters.     

Completely monotone multisequences, symmetric probabilities and a normal limit theorem

Let G _n,k be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc _n(β₁,β₂,…, β _k ), β₁,β₂,…, β_k = 0,1,2,…, β₁+β₂ + … +β _k ≤n,c _n(0,0,…, 0) = 1 and whenever β₀ ≤n - (β₁ + β₂ + … + β _k ) where Δc _n(β₁,β₂,…, β _k ) =c _n(β₁ + 1, β₂,…, β _k )+c _n(β₁,β₂+1,…, β _k )+…+c _n (β₁,β₂,…, β _k +1) -c _n(β₁,β₂,…, β _k ). Further, let Π _n,k be the set of all symmetric probabilities on {0,1,2,…,k} ⁿ . We establish a one-to-one correspondence between the sets G _n,k and Π _n,k and use it to formulate and answer interesting questions about both. Assigning to G _n,k the uniform probability measure, we show that, asn→∞, any fixed section {it{c_n}(β₁,β₂,…, β _k ), 1 ≤ Σβ _i ≤m}, properly centered and normalized, is asymptotically multivariate normal. That is, converges weakly to MVN[0, Σ _m ]; the centering constantsc ₀(β₁, β₂,…, β _k ) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex inR ^k.     

Ideal einstein,conformally flat and semi-symmetric immersions

Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR ^m (ε) of constant sectional curvature ε satisfies a basic inequality δ(n ₁,…,n _k )≤c(n ₁,…,n _k )H ²+b(n ₁,…,n _k )ε, whereH is the mean curvature of the immersion, andc(n ₁,…,n _k ) andb(n ₁,…,n _k ) are constants depending only onn ₁,…,n _k ,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n ₁,…,n _k ). In this paper, we first prove that every ideal Einstein immersion satisfyingn≥n ₁+…+n _k +1 is totally geodesic, and that every ideal conformally flat immersion satisfyingn≥n ₁+…+n _k +2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms. The author was supported by the NSFC and RFDP.     

Weierstrass multiple points on algebraic curves and ramified coverings

Here we prove the following result on Weierstrass multiple points. Theorem:Fix integers k, g with k≥5 and g>4k. Then there exist a genus g, Riemann surface X and k points P ₁, …,P _k of X such that for all integers b ₁≥…≥b _k ≥0we have:

Additive latin transversals

We prove that for every odd primep, everyk≤p and every two subsets A={a ₁, …,a _k } andB={b ₁, …,b _k } of cardinalityk each ofZ _p , there is a permutationπ ∈S _k such that the sumsa _i +b _π(i) (inZ _p ) are pairwise distinct. This partially settles a question of Snevily. The proof is algebraic, and implies several related results as well. Research supported in part by a State of New Jersey grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.     

On the replications of certain multiplicative designs

Bounds on the number of row sums in ann×n, non-singular (0,1)-matrixA sarisfyingA ^tA=diag (k ₁-λ₁,…,k _n-λ_n),k _j>λ_j>0,λ₁=…=λ_e,λ_e+1=…=λ_n are obtained which extend previous results for such matrices.     

Edge Maximal Non-Bipartite Graphs Without Odd Cycles of Prescribed Lengths

 Let ?(n;3,5,…,2k+1) denote the class of non-bipartite graphs on n vertices having no odd cycle of length ≤2k+1. We prove that for every G∈?(n;3,5,…,2k+1) and characterize the extremal graphs. We also study the subclass ℋ(n;3,5,…,2k+1) consisting of the hamiltonian members of ?(n;3,5,…, 2k+1). For this subclass the above upper bound holds for odd n. For even n we establish the following sharp upper bound:

Whitney theorem of interpolatory type for k-monotone functions

One of the main results of this paper is the following Whitney theorem of interpolatory type for k-monotone functions (i.e., functions f such that divided differences f[x ₀,…, x _k ] are nonnegative for all choices of (k + 1) distinct points x ₀,…, x _k .